Universe created/started 14 billion years ago so we cannot see beyond that timeline.Cause of that we have a limit of things that we can see. Now we are not in the center of the universe so my question can be awkward but an observer in 12 billion light-years away would see the same "observable universe" in size perspective or maybe also in some other perspectives ?

1-The confusing part is when we go further we go past in space so 12 billion light years away means 12 billion years ago so in that time universe was not as big as it's now.In that case the answer would be no.

2-In the other hand since "we are not in the center" according to him we are 12 billion light years away so the size of the observable universe will be the same.

I believe 2 is the correct answer.

Also every moment the radius of the observable universe increases but it does not mean anything cause in the edges of the observable universe theres nothing but CMB radiation

You're correct on both counts. (it should be radius, not diameter, though)

Without seeing the larger context it's just a guess, but in the first case it looks like the author might not mean the radius of the observable universe, but the event horizon (distance from which light emitted NOW can no longer reach us in the future).
However, it would still be incorrect, as ~14 Glyr is the current Hubble radius (where recession equals the speed of light) and it is not the event horizon.

You're correct on both counts. (it should be radius, not diameter, though)

Without seeing the larger context it's just a guess, but in the first case it looks like the author might not mean the radius of the observable universe, but the event horizon (distance from which light emitted NOW can no longer reach us in the future).
However, it would still be incorrect, as ~14 Glyr is the current Hubble radius (where recession equals the speed of light) and it is not the event horizon.

I see thanks for the reply ..I think they didnt want to include the expension since it may be confusing, maybe they will correct it in next chapters.

I guess theres a distinction between "entire universe" and "observable universe".
If I say something like this, "There could be a galaxy 100 billion light-year away " Is this make sense ? Since there could be another universe and within that universe another galaxy ?

Distance is a tricky concept in General Relativity. It is possible to accurately state that the only light we have ever observed has traveled for less than about 14 billion years, but the expansion of the universe messes with the concept of distance quite a lot. You could accurately state that that light traveled 14 billion light years, as that is the distance of the path the light traveled.

The light of the CMB, the furthest light we can observe, was emitted from matter roughly 43 million light years away. For a good fraction of the history of the universe, the light rays that were moving in our direction were getting further away. More recently, the rate of expansion slowed enough that that light started to gain ground instead, eventually reaching us nearly 14 billion years after it was emitted.

Today, the matter that emitted that radiation lies at roughly 46.5 billion light years distance. That matter has long since passed our cosmic horizon, so we won't ever be able to receive light emitted today from those galaxies.

Also every moment the radius of the observable universe increases but it does not mean anything cause in the edges of the observable universe theres nothing but CMB radiation

That's not quite why this doesn't matter. If we didn't live in a universe with dark energy, then if we could live for billions of years we would be able to see galaxies further and further away form, grow, and merge with one another.

Dark energy puts a kibosh on this, because it creates a horizon. Light that comes from galaxies that are more than something like 16 billion light years today can never reach us, no matter how long we wait. Furthermore, as the universe expands, more and more galaxies will move beyond this horizon (the horizon itself will eventually grow to about 17 billion light years and stay there). Eventually, nothing that isn't in the same galaxy cluster will remain within our horizon. So in a very real sense, our universe will, over large periods of time get smaller.

Distance is a tricky concept in General Relativity. It is possible to accurately state that the only light we have ever observed has traveled for less than about 14 billion years, but the expansion of the universe messes with the concept of distance quite a lot. You could accurately state that that light traveled 14 billion light years, as that is the distance of the path the light traveled.

The light of the CMB, the furthest light we can observe, was emitted from matter roughly 43 million light years away. For a good fraction of the history of the universe, the light rays that were moving in our direction were getting further away. More recently, the rate of expansion slowed enough that that light started to gain ground instead, eventually reaching us nearly 14 billion years after it was emitted.

Today, the matter that emitted that radiation lies at roughly 46.5 billion light years distance. That matter has long since passed our cosmic horizon, so we won't ever be able to receive light emitted today from those galaxies.

That's not quite why this doesn't matter. If we didn't live in a universe with dark energy, then if we could live for billions of years we would be able to see galaxies further and further away form, grow, and merge with one another.

Dark energy puts a kibosh on this, because it creates a horizon. Light that comes from galaxies that are more than something like 16 billion light years today can never reach us, no matter how long we wait. Furthermore, as the universe expands, more and more galaxies will move beyond this horizon (the horizon itself will eventually grow to about 17 billion light years and stay there). Eventually, nothing that isn't in the same galaxy cluster will remain within our horizon. So in a very real sense, our universe will, over large periods of time get smaller.

How can be the horizon radius 17 Billion light- year since its now 46.5 billion.Did you mean that the galaxies further then 16-17 billion light-year away galaxies will be no longer visible ?

I generally understood your idea.Since theres dark energy and galaxies move away further from us we will be not able to see them after a time.

Distance is a tricky concept in General Relativity. It is possible to accurately state that the only light we have ever observed has traveled for less than about 14 billion years, but the expansion of the universe messes with the concept of distance quite a lot. You could accurately state that that light traveled 14 billion light years, as that is the distance of the path the light traveled.

The light of the CMB, the furthest light we can observe, was emitted from matter roughly 43 million light years away. For a good fraction of the history of the universe, the light rays that were moving in our direction were getting further away. More recently, the rate of expansion slowed enough that that light started to gain ground instead, eventually reaching us nearly 14 billion years after it was emitted.

Today, the matter that emitted that radiation lies at roughly 46.5 billion light years distance. That matter has long since passed our cosmic horizon, so we won't ever be able to receive light emitted today from those galaxies.

That's not quite why this doesn't matter. If we didn't live in a universe with dark energy, then if we could live for billions of years we would be able to see galaxies further and further away form, grow, and merge with one another.

Dark energy puts a kibosh on this, because it creates a horizon. Light that comes from galaxies that are more than something like 16 billion light years today can never reach us, no matter how long we wait. Furthermore, as the universe expands, more and more galaxies will move beyond this horizon (the horizon itself will eventually grow to about 17 billion light years and stay there). Eventually, nothing that isn't in the same galaxy cluster will remain within our horizon. So in a very real sense, our universe will, over large periods of time get smaller.

Excellent post, very detailed explanation.

I do have one question about this part "More recently, the rate of expansion slowed enough that that light started to gain ground instead"
As I understand the current theory, the rapid rate of expansion of the early universe did slow down some 5 billion years ago, but we now see evidence it has sped up again due to dark energy. Is this your understanding also?

I do have one question about this part "More recently, the rate of expansion slowed enough that that light started to gain ground instead"
As I understand the current theory, the rapid rate of expansion of the early universe did slow down some 5 billion years ago, but we now see evidence it has sped up again due to dark energy. Is this your understanding also?

The rate of expansion in kimbyd's post is another name for the Hubble parameter ##H(t)## - it tells you by what fraction all distances grow at a given time.
For example, its value at the current epoch ##H_0## (aka Hubble constant) netts approximately 1/144 % of growth per million years (which is just another way of writing the more familiar ~68 km/s/Mpc).
The Hubble parameter has always been and will continue to decrease, in the far future asymptotically approaching some fixed value determined by dark energy.

What used to be decelerating and has since started to accelerate is the growth of the scale factor ##a##. The scale factor tells you how large all distances are compared to those same distances at some other time. It's like the nett growth, as opposed to ##H## being instantaneous percentage rate.

A good analogy to visualise the difference is a savings account in a bank. The scale factor is how much money you have. The Hubble parameter is the monthly interest rate.
If the interest rate does not change between months, the amount of savings will grow exponentially. If the rate goes down eventually reaching zero, the amount of savings will eventually stop growing. If the interest rate goes down but to some positive value*, the savings may at first grow by less every month, but eventually will start growing exponentially (i.e. there will be a change from decelerated to accelerated growth).

*Example of such a rate: 1+1/n %/month, where n is the number of months the money's been sitting on the account = the rate approaches 1 % as time passes.

The rate of expansion in kimbyd's post is another name for the Hubble parameter ##H(t)## - it tells you by what fraction all distances grow at a given time.
For example, its value at the current epoch ##H_0## (aka Hubble constant) netts approximately 1/144 % of growth per million years (which is just another way of writing the more familiar ~68 km/s/Mpc).
The Hubble parameter has always been and will continue to decrease, in the far future asymptotically approaching some fixed value determined by dark energy.

What used to be decelerating and has since started to accelerate is the growth of the scale factor ##a##. The scale factor tells you how large all distances are compared to those same distances at some other time. It's like the nett growth, as opposed to ##H## being instantaneous percentage rate.

A good analogy to visualise the difference is a savings account in a bank. The scale factor is how much money you have. The Hubble parameter is the monthly interest rate.
If the interest rate does not change between months, the amount of savings will grow exponentially. If the rate goes down eventually reaching zero, the amount of savings will eventually stop growing. If the interest rate goes down but to some positive value*, the savings may at first grow by less every month, but eventually will start growing exponentially (i.e. there will be a change from decelerated to accelerated growth).

*Example of such a rate: 1+1/n %/month, where n is the number of months the money's been sitting on the account = the rate approaches 1 % as time passes.

I'm not sure I understand this. Is the rate of expansion of the universe slowing down or speeding up, or do we know?

I thought the rate of ACCELERATION was slowing down, meaning the rate of expansion is continuing to increase and will continue to continue to increase.

I'm not sure what would that mean, but it's likely a matter of using different names for the same thing.

Here's the standard usage:

##a(t)## - scale factor, 'size of the universe' as compared to today (not the observable universe - the distance between any two arbitrarily chosen points in the universe, e.g. some two galaxies)
##\dot a > 0## - size of the universe is getting bigger over time, 'expansion of the universe'
##\dot a < 0## - size of the universe is getting smaller over time, 'contraction of the universe'

##\frac{\dot a}{a} = H## - Hubble parameter, 'rate of expansion' = by what fraction of its size does the universe grow per unit time
##\dot H > 0## - rate of expansion is increasing over time, 'accelerated rate of expansion'
##\dot H < 0## - rate of expansion is decreasing over time, 'decelerated rate of expansion'

##\ddot a > 0## - growth of the universe is speeding up over time, 'accelerated expansion'
##\ddot a < 0## - growth of the universe is slowing down over time, 'decelerated expansion'

I'm not sure I understand this. Is the rate of expansion of the universe slowing down or speeding up, or do we know?

Can you tell me where have I lost you with the interest rate analogy? Can you see how you can gain more money in interest (expansion accelerating) even though the interest (expansion) rate is going down?

Can you tell me where have I lost you with the interest rate analogy? Can you see how you can gain more money in interest (expansion accelerating) even though the interest (expansion) rate is going down?

It's terminology. We're saying the same thing in terms of what's happening. I look at the interest as the rate of acceleration and the overall amount of money as the expansion, so I see the acceleration (interest) going down slightly but the amount of money (expansion) continuing to increase more and more because of compound interest.